Conformal Mappings and Loewner Evoluation

共形映射和 Loewner 演化

• 批准号: 0201435
• 负责人: Donald Marshall
• 金额: $ 10.8万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0201435&HistoricalAwards=false
• 关键词: Conformal; Mappings; Loewner; Evoluation

Proposal Number: DMS-0201435PI: Donald Marshall and Steffen RohdeABSTRACTConformal mapping and Loewner evolutionsMarshall and Rohde will investigate conformal mappings generated by the Loewner differential equation, and related topics. TheLoewner differential equation describes the flow associated withthe conformal mappings onto a continuously decreasing sequence ofsimply connected planar domains. It relates a sequence of domainsto a real-valued function, the driving term of the equation.Schramm's recent discovery of the stochastic Loewner evolutionSLE, the Loewner equation driven by one-dimensional Brownianmotion, has opened up a new area of investigations involvingconformal mappings, probability theory and mathematical physics.The Loewner equation is also related to an algorithm fornumerical conformal mapping.Conformal mappings have applications in many areas, both within and outside of mathematics, such as control theory, heatconduction, fluid dynamics, and complex dynamics. They are oftenused to change coordinates from one region to a simpler regionlike a disc. Regions with smooth boundaries are well understood.However, the appearance of fractals in many branches of science led to the natural problem of investigating regions bounded by highly nonsmooth, fractal curves, from the conformal mappingpoint of view. In recent years, fractal curves generated byrandom processes arising, for instance, in statistical physics,have received enormous attention. The core of Marshall's andRohde's research is to better understand random fractal curvesby means of conformal mappings, and conversely to study someproblems about conformal mappings by analyzing domains boundedby fractal curves.

提案编号：DMS-0201435 PI：Donald马歇尔和Steffen Rohde摘要保形映射和Loewner演化马歇尔和Rohde将研究由Loewner微分方程生成的保形映射以及相关主题。Loewner微分方程描述了平面单连通域上连续递减序列的共形映射流。Schramm最近发现的随机Loewner演化SLE，即由一维布朗运动驱动的Loewner方程，开辟了一个新的研究领域，涉及共形映射，Loewner方程也与数值保角映射的算法有关。保角映射在许多领域都有应用。数学领域，包括数学内部和外部，如控制理论，热传导，流体动力学和复杂动力学。它们通常用于将坐标从一个区域改变到一个更简单的区域，如圆盘。具有光滑边界的区域是很好理解的。然而，分形在许多科学分支中的出现导致了从保角映射的观点来研究由高度非光滑的分形曲线包围的区域的自然问题。近年来，由随机过程产生的分形曲线，例如，在统计物理学中，受到了极大的关注。马歇尔和罗德研究的核心是利用共形映射来更好地理解随机分形曲线，反过来又通过分析分形曲线所围区域来研究共形映射的一些问题。